There are many physical problems that are characterized by sudden changes in their states.
These sudden changes are known as impulsive effects in the system. Impulsive differential equations are the differential equations involving impulse effect. It has significants applications in several real word problems namely mechanical systems with impact, biological systems such as heart beats, blood flow, population dynamics, ecology, control theory, medicine and so on. There are many books available on impulsive differential equations e. g. Bainov and Simeonov 1, Lakshmikantham 2 and Benchohra 3.
In the existing literature, there are two types of impulsive systems. One is instantaneous impulsive system and other one is non-instantaneous impulsive systems. In instantaneous impulsive system, the duration of these sudden changes is very small in comparison with the duration of an entire evolution process as like in shocks and natural disasters and in noninstantaneous impulses, the duration of these changes continue over a finite time interval. A very well known application of non-instantaneous impulses is the introduction of insulin in the bloodstream which is the abrupt change and the consequent absorption which is a gradual process as it remains active for a finite time interval. For the initial studies related with the existence, uniqueness, controllability and stability of non-instantaneous impulsivesystems, we can find in 4–8. In 1960, Kalman 9–11 was the first person who introduced the concept of controllability and observability which formed the backbone of modern control theory. Roughly speaking, controllability means the steering of a dynamical control system from an initial state to the desired final state by using a suitable control available in the system and by the observability we means, that we can uniquely find the initial state with the help of corresponding system input and output.
These results have been studied in depth in both the continuous and discrete cases. In 1988, Hilger 12 introduced the calculus on time scales in his Ph.D thesis. The study of dynamic equations encapsulates both the continuous as well as discrete analysis of the system.
Since its inception, study of dynamical systems on time scales has gained a great deal of international attention and many researchers have found the applications of time scales in heat transfer system, population dynamics and as well as in economics. We will give more details about time scales in the next section.